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Grade 11 · lesson 4 of 4

Multiply a Complex Number by Its Conjugate

A conjugate pair differs only in its imaginary-part sign.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Evaluate (2+i)(2−i), using i²=−1. Candidate answers: (A) 10 (B) 4 (C) 5 (D) 6.
  • Independent check inputs: Evaluate (3+i)(3−i), using i²=−1. Candidate answers: (A) 10 (B) 11 (C) 20 (D) 9.

Learn the idea

Root expressions can be simplified through perfect-power factors and rational exponents. Squaring equations can introduce extraneous candidates, while complex arithmetic extends the number system using an imaginary unit with square negative one. A conjugate pair differs only in its imaginary-part sign. Multiplying cancels the cross terms, and replacing i² with negative one makes the resulting value real.

Here is the method on the supplied teaching case. Distribute: 2²−2i+2i−i². The imaginary terms cancel, and −i²=1, so the result is 4+1=5.

Treat i as a symbol with the specific rule i²=−1. The subtraction of i² becomes addition of one; multiplying only real parts would omit the imaginary-square contribution.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Evaluate (2+i)(2−i), using i²=−1. Candidate answers: (A) 10 (B) 4 (C) 5 (D) 6. Solution: Distribute: 2²−2i+2i−i². The imaginary terms cancel, and −i²=1, so the result is 4+1=5. The final answer is 5. Treat i as a symbol with the specific rule i²=−1. The subtraction of i² becomes addition of one; multiplying only real parts would omit the imaginary-square contribution.

Your turn

Check your reasoning

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Helpful tips
  • A conjugate pair differs only in its imaginary-part sign.
  • Treat i as a symbol with the specific rule i²=−1.
Evaluate (3+i)(3−i), using i²=−1.

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See the answer and explanation

10

Distribute: 3²−3i+3i−i². The imaginary terms cancel, and −i²=1, so the result is 9+1=10.

Take it further

Changed-input task: Evaluate (9+i)(9−i), using i²=−1. Candidate answers: (A) 164 (B) 81 (C) 82 (D) 83. Show the mathematical reason, not only the selected answer. Expected result: 82. Distribute: 9²−9i+9i−i². The imaginary terms cancel, and −i²=1, so the result is 81+1=82.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Evaluate (4+i)(4−i), using i²=−1.
Need a hint?
  • A conjugate pair differs only in its imaginary-part sign.
  • Treat i as a symbol with the specific rule i²=−1.
2. Evaluate (5+i)(5−i), using i²=−1.
Need a hint?
  • A conjugate pair differs only in its imaginary-part sign.
  • Treat i as a symbol with the specific rule i²=−1.
3. Evaluate (6+i)(6−i), using i²=−1.
Need a hint?
  • A conjugate pair differs only in its imaginary-part sign.
  • Treat i as a symbol with the specific rule i²=−1.

Your choices are temporary. Feedback appears after you check all the answers.

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